Observer/Kalman-Filter Time-Varying System Identification - AIAA

892 MAJJI, JUANG, AND JUNKINS
P k 1
:
2
h o 3 2
3
k 1;k
C k 1 G k
h o k 2;k
C k 2 A k 1 G k
6 . 7 6
.
7
4 . 5 4
.
5
h o k m;k C k m ;A k m 1 ; ... ;A k 1 G k
2
3
C k 1
C k 2 A k 1
6
.
7
4
5 G k O m
k 1 G k (37)
C k m ;A k m 1 ; ... ;A k 1
such that a least-squares solution for the gain matrix at each time step
is given by
G k O m †
k 1 P k 1 (38)
However, from the developments of the companion paper, we find
that it is indeed impossible to determine the observability Gramian
in the true coordinate system [9], as suggested by Eq. (38). The
computed observability Gramian is, in general, in a time-varying and
unknown coordinate system denoted by O m
k 1 at the time step t k 1.
We will now show that the gain computed from this time-varying
observability Gramian (computed) will be consistent with the timevarying
coordinates of the plant model computed by the TVERA
presented in the companion paper. Therefore, upon using the
computed observability Gramian (in its own time-varying coordinate
system) and proceeding with the gain calculation, as indicated by
Eq. (38), we arrive at a consistent computed gain matrix. Given a
transformation matrix T k 1 ,
P k 1 O m
k 1 G k O m
k 1 T k 1Tk
1
such that
^G k Tk
1
1 G k
1 G k
^O m
k 1 T k
1
1 G k
^O m ^G k 1 k
(39)
^O m
k 1 † P k 1 (40)
therefore, with no explicit intervention by the analyst, the realized
gains are automatically in the right coordinate system for producing
the appropriate TOKID closed loop. For consistency, it is often
convenient if one obtains the first few time-step models, as included
in the developments of the companion paper. This automatically
gives the observability Gramians for the first few time steps to
calculate the corresponding observer gain matrix values. To see that
the gain sequence computed by the algorithm is, indeed, in consistent
coordinate systems, recall the identified system and control influence
and the measurement sensitivity matrices in the time-varying
coordinate systems, to be derived as (refer to companion paper [9])
^A k T 1
k 1 A kT k ; ^B k T 1
k 1 B k; ^C k C k T k (41)
The TOKID closed-loop system matrix, with the realized gain matrix
sequence, is seen to be consistently given as
V. Relationship Between the Identified Observer
and a Kalman Filter
We now qualitatively discuss several features of the observer,
realized from the algorithm presented in the paper. Constructing the
closed loop of the observer dynamics, it can be found to be
asymptotically stable, as purported at the design stage. Following the
developments of the time-invariant OKID paper, we use the wellunderstood
time-varying Kalman-filter theory to make some intuitive
observations. These observations help us address the important issue:
variable-order GTV-ARX model fitting of I/O data and what it all
means. Insight is also obtained as to what happens in the presence of
measurement noise. In the practical situation, for which there is the
presence of the process and the measurement noise in the data, the
GTV-ARX model becomes a moving average model that can be
termed as the GTV autoregressive moving average with exogenous
input (GTV-ARMAX) model (generalized is used to indicate
variable order at each time step). A detailed quantitative examination
of this situation is beyond the scope of the current paper: the authors
limit the discussions to qualitative relations.
The Kalman-filter equations for a truth model given in Eq. (A1) are
given by
or
^x k 1 A k ^x k B k u k (43)
^x k 1 A k I K k C k ^x k B k u k A k K k y k (44)
together with the propagated output equation
^y k C k ^x k D k u k (45)
where the gain K k is optimal [see expression in Eq. (A16)]. As
documented in the standard estimation theory textbooks, optimality
translates to any one of the equivalent necessary conditions of
minimum variance, maximum likelihood, orthogonality, or Bayesian
schemes. A brief review of the expressions for the optimal gain
sequence is derived in the Appendix A, which also provides an
insight into the useful notion of orthogonality of the discrete
innovations process in addition to deriving an expression for the
optimal gain matrix sequence [see Eq. (A16) for an expression for the
optimal gain]. From an I/O standpoint, the innovations approach
provides the most insight for analysis and is used in this section.
Using the definition of the innovations process " k
: y k ^y k , the
measurement equation of the estimator [shown in Eq. (45)] can be
written in favor of the system outputs, as given by
y k C k ^x k D k u k " k (46)
Rearranging the state propagation shown in Eq. (43), we arrive at a
form given by
^x k 1 A k I K k C k ^x k B k u k A k K k y k
~A k ^x k
~B k v k
(47)
^A k
^G k ^C k T 1
k 1 A k G k C k T k (42)
in a kinematically similar fashion to the true TOKID closed loop. The
nature of the computed stabilizing (deadbeat or near-deadbeat) gain
sequence is best viewed from a reference coordinate system, as
opposed to the time-varying coordinate systems computed by the
algorithm. The projection-based transformations can be used for this
purpose and are discussed in detail in the companion paper [9].
with the definitions
~A k A k I K k C k ; ~B k B k A k K k ; v k
u k
y k
(48)
Notice the structural similarity in the layout of the rearranged
equations to the TOKID equations in Sec. III. This rearrangement
helps in making comparisons and observations as to the conditions in
which we actually manage to obtain the Kalman-filter gain sequence
(or otherwise).
Starting from the initial condition, the I/O relation of the Kalmanfilter
equations can be written as